The robustness of power grids is a central topic in the design of the so called "smart grid". As the total amount of load in the power grid is becoming more and more each day, and the sizes of power grids are becoming larger and larger each year, it will result in larger damages to the society if the power grids broke down. For example, the Northeast blackout of 2003 in the US resulted in a loss of 61,800 MW of electric load that served more than 50 million people ELCON (2003). In this paper we try to analyze the measures of importance of the nodes in a power grid under cascading failure. Moreover, we introduce the notion of degree correlation for the case where only the statistical information of a power grid is known. Mean field theory is used for our analysis. With these efforts, we can distinguish the most vulnerable nodes and protect them, improving the safety of the power grid. Also we can measure if a structure is proper for power grids. Above all, a new multi-loop structure for smart grid is shown to be better than the contemporary one.
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